Believe it or not, it is very possible to catch Mewtwo in Pokémon Fire Red and Leaf Green without cheating! He's just sitting there in Cerulean Cave. Actually, catching him especially without a Master Ball is tricky.

Steps

1

Before you ever attempt to catch him, save the game.

2

Load up on Ultra Balls and Timer Balls (found on Island 2).

3

Don't fight Mewtwo! It helps but wastes time.

4

Use your Dark Pokémon out front, or put him to sleep constantly.

5

Throw, throw, throw. Once you have thrown or taken up 30 turns, and if you haven't already caught him, switch over to the Timer Balls and use those.

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Tips

Dark Pokémon are more useful than putting him to sleep.

If you have a Master Ball, Mewtwo will 100% get caught, and you don't have to waste time. However, if you haven't caught Entei, Raikou, or Suicune yet, you may want to use it on them instead, because they roam around Kanto rather than staying in one place.